Let \(R_{\exp } = (R,+,\cdot ,\exp )\) . Wilkie proved that \(R_{\exp }\)  is
model complete [1]}. As a direct consequence of this theorem, each definable sets in \(R_{\exp }\)  is the image of the zero set of a function in \(P(x,y,e^x,e^y)=0\)  under a natural projection (see page 3 of [2]} or [3]}).
Then by a Khovanskii result on fewnomials [4]}, \(R_{\exp }\)  is an o-minimal structure. An analytic proof of Wilkie’s theorem is given in [5]}.
